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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import OAI.Probability.InvariantIsing.Core.BoundedObservableMinimum
import OAI.Probability.IsingPerceptron.EnrichedCGF
namespace OAI
/-! A bounded random observable has a differentiable disorder-averaged CGF.
This applies to the actual temperature observable retaining common disorder. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
lemma integrable_bounded_random_tilted_mean {Ω X : Type*}
[MeasurableSpace Ω] [MeasurableSpace X]
(P : Measure Ω) [IsProbabilityMeasure P] (ν : Ω → Measure X) (hν : Measurable ν)
[∀ ω, IsProbabilityMeasure (ν ω)] (Y : Ω → X → ℝ)
(hY : Measurable (Function.uncurry Y)) (B : ℝ) (hB : ∀ ω x, |Y ω x| ≤ B) (t : ℝ) :
Integrable (fun ω => ∫ x, Y ω x ∂(ν ω).tilted (fun x => t * Y ω x)) P := by
have hm := measurable_random_tilted_integral hν (hY.const_mul t) hY
apply Integrable.of_bound hm.aestronglyMeasurable B
exact ae_of_all _ fun ω => by
simpa only [Real.norm_eq_abs, Function.comp_apply, Function.uncurry_apply_pair, id_eq] using bounded_tilted_integral_abs (ν ω)
(hY.comp (measurable_const.prodMk measurable_id)) (hB ω)
(hY.comp (measurable_const.prodMk measurable_id)) (hB ω) t
theorem hasDerivAt_bounded_random_cgf {Ω X : Type*}
[MeasurableSpace Ω] [MeasurableSpace X]
(P : Measure Ω) [IsProbabilityMeasure P] (ν : Ω → Measure X) (hν : Measurable ν)
[∀ ω, IsProbabilityMeasure (ν ω)] (Y : Ω → X → ℝ)
(hY : Measurable (Function.uncurry Y)) (B : ℝ) (hB : ∀ ω x, |Y ω x| ≤ B) (t : ℝ) :
HasDerivAt (fun s => ∫ ω, cgf (Y ω) (ν ω) s ∂P)
(∫ ω, ∫ x, Y ω x ∂(ν ω).tilted (fun x => t * Y ω x) ∂P) t := by
apply hasDerivAt_mean_cgf ν Y
· exact ae_of_all _ fun ω s => exp_mul_integrable_of_bound (ν ω)
(hY.comp (measurable_const.prodMk measurable_id)) (hB ω) s
· exact fun s => integrable_bounded_random_cgf P ν hν (Function.uncurry Y) hY B hB s
· exact fun s => integrable_bounded_random_tilted_mean P ν hν Y hY B hB s
end InvariantIsing
end
end OAI
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